Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest, tacle reverse percentage prolems and negotiate cominations of increases and decreases consider negative and fractional powers and indices calculate direct and inverse variation identify sequences where part of the formula is n Context In this section of the course, we tacle some more advanced topics. Often they are an extension of material you have already covered earlier in the course. You may need to remind yourself of the sills learnt in Lessons One, Four, Ten and Thirteen. Oxford Open Learning
Lesson Woring with Irrational Numers As you now, an irrational numer is one which cannot e expressed as a fraction, e.g. the square root of, written as. A calculator will show it in decimal form as.7008 i.e. it never terminates or repeats itself. (pi) is another example of an irrational numer. =.9.. A surd is an irrational numer that contains a square root, e.g. 7 0 - + etc Sometimes surds can e simplified. There are two rules which enale simplification to tae place. Consider, for example, 7. There are two operations involved here: firstly, square rooting each numer, and secondly, multiplying. Now it turns out that these two operations can e performed in the reverse order, so that the two numers are first multiplied and then the square of the result is found: 7 7 If you try oth versions on a calculator you should get the same result:.8769 The strategy for simplifying a surd can e summarised as: Express the numer under the root sign as a product of two numers. This is called a factor pair, one of which is a square numer. You are looing for the largest square numer factor. Use the aove rule to reverse the order of the two operations. Log on to Twig and loo at the film titled: Irrational Numers www.ool.co.u/76gc Beside the now-famous theorem which ears his name, discover the izarre life of Pythagoras and his Brotherhood, in which maths was the language of the Gods and irrational numers were heretical.
Module 9: Further Numers and Equations Example Express each of the following as the simplest possile surd : (a) 8 () 7 (c) 8 The aim is to express 8 as the product of two numers, one of which is a square numer. In fact, the only way to split 8 into a product is as, and we see that one of the numers () is a square numer. So 8. Reversing the order of operations, we get. But we now that is simply. We can therefore conclude that 8 can e written as, although it is usually written as. (When no operation is indicated, it must e multiplication.) If you chec on a calculator, you should find that and 8 are oth.887 Try dividing 7 y each of the square numers in turn, until one goes exactly without remainder. We do not other with the first square numer,. There are remainders when 7 is divided y, 9 and 6. However, there is no remainder when dividing 7 y : 7. So 7 or even more simply as. 8 is divisile y the second square numer,. So we could proceed to simplify 8 as. However, it is much more efficient to find the largest square numer which divides into 8. In fact, the fourth square numer (6) divides into 8. So 8 6 6 or simply. Activity (a) Express each of the following as the simplest possile surd : 8 () 0 (c) 7 (d) (e) 7 (f) 80 (g) 98 (h) 6
Lesson Surds: Rationalising the Denominator We have already looed at ways of rationalizing (or simplifying) the denominator of a fraction. A surd is an irrational root or an expression containing irrational numers or complex numers, e.g. surd. + is a Consider the fraction. Is there any way of simplifying it? Often it is helpful to rationalize the denominator, i.e. turn the denominator into a rational numer. How do we do that? By multiplying top and ottom y a quantity which would enale us to eliminate the square root sign. In this example, we can multiply top and ottom y : = = Activity Rationalize the denominator in the following: (a) () 0 (c) Reverse Percentage Prolems We now extend the idea of growth rate, although it would e etter to use the more general term multiplier since we are aout to consider things that reduce. Recall the rule for finding the multiplier when something increases y %: Convert % into a decimal: % = = 0.0 00
Module 9: Further Numers and Equations Add the decimal to : + 0.0 =.0 Now, if something were to reduce y %, then the only difference is in the second line: instead of adding the decimal, we sutract the decimal. So to reduce something y %, the quic way is to multiply y 0.9 (since 0.0 = 0.9). Mae sure you understand the following tale. Multiplier for: % an increase A decrease.0 0.98..0 0.99. 0.8 6.9.69 0.6 Example The multiplier for repeated reductions can e used in the same way as for an increase. Depreciation is one such context. If something depreciates at 0%, then it loses 0% of its value each year. This is lie compound interest in reverse, so the same technique applies: Final value = original value multiplier years This powerful formula wors for oth increases and decreases. A car depreciates y % each year. If it was worth 000 when new, what will it e worth after five years? Reverse Calculations Convert % to a fraction: % = 0.. Now sutract 00 from for a reduction: 0. = 0.8. The value after five years will e 000 0.8.6. We are sometimes required to wor acwards : given the value after an increase/decrease find the original value. Loo again at the formula for otaining the final value after an increase/decrease: Final value = original value multiplier years The ey detail is the multiplication sign. To reverse the calculation we perform the opposite of multiplication: division. Tale concerns four situations which are all forward, we are given the original amounts, and need to find the final amount after a percentage increase or decrease. Notice that the multipliers are greater than one for the increases, and less than one for the decreases.
Lesson Tale Original % Increase/ Multiplier Final amount amount decrease? 00 Increase.0 00. 0 = 0 00 7. Decrease 0.9 00 0.9 = 90.0 80. Increase.0 80.0 = 9.0 96 Decrease 0.97 96 0.97 = 96.8 In contrast, Tale concerns situations which are all acward. We are given the final amount after a certain percentage increase or decrease, and we need to wor acward to find the original amount. Again, notice that the multipliers are greater than one for the increases, and less than one for the decreases. Tale Final amount % Increase or decrease? Multiplier Original amount 7. Increase.0 7.0 = 70. 960 6.8 Decrease 0.9 960 0.9 = 00.0.78 9. Increase.09.78.09 = 06.6 79.07. Decrease 0.868 79.07 0.868 = 009.09 Concentrate on the one important difference etween Tales and. In Tale the calculations in the right hand columns are all multiplications (ecause we are woring forwards ); in Tale they are all divisions (we are woring acwards ). Cominations of Increases and Decreases Example An investment increases in value y % in the first year, 0.% in the second year, and decreases y.% in the third year. What is the value of the investment at the end of the three years if the initial amount was 000? The multipliers for the three years are.0,.00 and 0.98 respectively. We simply multiply the original amount y each of the multipliers: 000.0.00 0.98 = 09.6 Activity Wor out the following. You may use a calculator: 6
Module 9: Further Numers and Equations. A sum is invested in a an at a fixed rate of 0% annual compound interest. After years, the sum has grown to,7.69. What was the original sum?. A sum is invested in a an at a fixed rate of 6% annual compound interest. After years, the sum has grown to.6. What was the original sum? Activity. Copy and complete the following tale: Original amount % Increase or decrease? Multiplier Final amount a 00.7 Increase 680.8 Decrease c 79.6 Increase d 000 8.7 Decrease e.9 Increase 6. f. Decrease 709. g.8 Increase 7.80 h.6 Decrease 9.7. A special investment account pays a fixed compound interest of 7% per annum. Money can neither e added nor withdrawn during the first three years of the account. a) If 00 is paid in to the account at the eginning, how much will it e worth at the end of the first three years? ) Suppose you need the account to e worth 000 at the end of the first three years. What must e the value of your initial deposit?. A new car costs 990. It depreciates y % in the first year, 0% in the second year and % in the third year. How much will the car e worth after the three years?. An investment increases in value y 8% in its first year. It decreases in value y 8% in the second year. If the initial value of the investment is 000, what is the value of the investment after the two years? Are you surprised?. An investment increases in value y % in its first year. It decreases in value y 0% in the second year. If the initial value of the investment is 000, what is the value of the investment after the two years? Are you still surprised? 7
Lesson Negative and Fractional Powers You should recall how to find square roots and cue roots. We now introduce some new notation. In general, the new notation is summed up as follows: a n n a This simply means that if a power is a fraction with numerator one, then this means the nth root of the numer. Just get used to the new notation y looing carefully at the following examples. The calculations are revision (i.e. changing the root form into the simplest possile form ). Concentrate particularly on the equivalence of the first two columns which say the same thing in two different ways. Fractional Index form 9 8 6 00000 Root form Simplest possile form 9 8 6 00000 0 Note that if there is no little numer to the left of the root sign, then this means a square root: in other words, a two is implied. So, for example, 9 means 9. We now need to consider the case when the numerator of a fractional index is not one. We will need a technique from Lesson One. One of the Index Laws said that if there are powers inside and outside a racet, then these powers can e multiplied. For instance: Consider, for example, 8. We can thin of the fraction as. This may seem a strange thing to do, ut it enales us to use the aove Index Law: 8
Module 9: Further Numers and Equations 8 8 8 But we already now that 8 means the cue root of 8, and has the value. We can therefore do 8 8 8. Let us reflect on the parts that the numerator and denominator of the fraction have played. The denominator is three, and we found a cue root. The numerator is two and we found a square (i.e. the power two). This is always the case. It turns out that we could even reverse the order of the two operations. If we returned, for example, to the case of 8, ut squared first, we would otain 6. If we calculated the cue root of 6 we would otain, the same as efore. However, it is usually much easier to perform the root first. Consider the following Example. Example Simplify 6 The numerator of the fraction is three: this means that we need the cue (i.e. a power three). The denominator of the fraction is four: this means that we need a fourth root. If we do the fourth root first, we otain, (since 6 ). Now find the cue of, which is 8. However, loo what happens if we reverse the order of the two operations. Do the cue first: 6 096. We now need to find the fourth root of 096. You need to now simple roots of small numers. You should e reassured that you are not expected to carry with you at all times the information that the fourth root of 096 is 8. Of course, a calculator would help. However, you need to e ale to simplify expressions lie that in Example without a calculator. So, to sum up, it is wiser to perform the root first and the power afterwards: this usually maes the mental arithmetic much easier. 9
Lesson Negative Fractional Indices Example Rememer that a negative power means that we mentally remove the minus sign and then do the reciprocal of the result. This also wors for negative fractional indices, if we eep calm and proceed one step at a time. Simplify Mentally remove the minus sign:. The five means that we need the fifth root of, which is (since ). The three means that we need the cue of, which is 8. Now we need to do the reciprocal of, in other words. The ottom of the fraction is 8, so our final answer is 8. Fractional Indices on a Calculator Example This deserves a separate section. There are various ways of finding a fractional power of a numer on a modern scientific calculator. It is suggested that you try all the following techniques. However, then decide on a method that you lie, that you will rememer and that wors relialy for you. Use a calculator to evaluate 9 Principal Method Exact eystroe sequence Answer y Power utton: x y 9 x c 7 ( a ) = c Fraction utton: a y Power utton: x Division 9 y x ( ) = 7 Root utton: x x 9 = 7 Example Note that the racets in oth of the first two methods are essential. If you omit these racets, you get a different answer (which is incorrect). Simplify 6 0
Module 9: Further Numers and Equations Principal Method Exact eystroe sequence Answer y y Power utton: x c 6 x 8 ( a ) = Fraction utton: Power utton: Division y x c a y 6 x ( ) = y y Power utton: x 6 x 0.7 = 8 8 Again, note that the racets in the first two methods are essential. The third method is not always availale. For 7 instance, if we needed to find 8, then we cannot replace the fraction y a decimal. This is ecause the decimal 7 version of 7 recurs with a repeating loc of six digits: 0.8787 If we try and use a decimal version of 7, it will only e an approximation. Fractional Powers of a Fraction The aove ideas can e extended to find a fractional power of a fraction. This is largely revision. Example Simplify: (a) 6 8 () 6 8 (a) As we did earlier, reverse the order of the two operations, i.e. do the powers first and then the division: 6 6 8 8 Recall that the four means a fourth root and the three means a cue. 6 6 8, and similarly 8 8 7 8 The final answer is therefore. 7.
Lesson () The only difference etween (a) and () is the minus sign. The minus sign means that we do the reciprocal of the answer for (a). Recall that to do the reciprocal of a fraction, we swap the top and ottom. The answer to () is therefore 8 7, which we could also write as the mixed numer. 8 Finally, a reminder that it is often easier to wor with improper fractions rather than mixed numers. Example 6 Simplify: (a) () (a) Change oth the mixed numers into improper fractions: 9 and. So 9. Reverse the order of the two operations (power and division): 9. The two means square root, and the three means cue: 9 8 7. We could rewrite this final answer as if required. 8 () The minus sign means do the reciprocal. It is est to wor 7 8 with the improper fraction. The reciprocal of this is. 8 7 Note that it is possile to doule-chec the answers to Example 6 using a calculator, if you are careful. However, your calculator may require some persuasion. c c ( a a ) y x ( c c a a ) =
Module 9: Further Numers and Equations The racets are again essential. Also, the aove sequence performed on a modern Casio generates a decimal answer. So you need to persuade the calculator to convert the decimal into a fraction y then pressing the fraction utton a again. This gives the answer in the form of the mixed numer. If 8 required, we could then as the calculator to convert this to the improper fraction 8 7 y pressing SHIFT followed y c c a. Activity. Copy and complete the following tale. Fractional Index form 6 00 (?) Root form Simplest possile form 6 7??. Without using a calculator, simplify the following: (a) 6 () 6 (c) (d) (e) 7 (f) 7. Without using a calculator, simplify the following: (a) 6 () 6 (c) (d) (e) 0000 (f) 0000
Lesson. Use a calculator to simplify the following (write each answer as a fraction): (a) 0076 () 7776 (c) 096 (d) 9 (e) 0. Without using a calculator, simplify the following (write each answer as a fraction): 8 (a) 7 8 () 7 8 (c) 7 8 (d) 7 6. Without using a calculator, simplify the following (write each answer as a fraction): (a) 6 () 6 7. Use a calculator to simplify the following (write each answer as a fraction): 6 (a) 9 () Proportionality The proaility of lung cancer is proportional to how much you smoe. The amount of radioactivity in a piece of material is inversely proportional to the time. It is the jo of mathematicians and statisticians to analyse and use statements such as these. Many of these prolems are complex ut we will only loo at the simple asic techniques. Direct Variation and Inverse Variation We sometimes say that two quantities, such as x and y, are in direct proportion. The mathematical meaning is the same as the everyday meaning. If one quantity doules, for instance, then so does the other. If one quantity ecomes halved, then so does the other. Inverse proportion is a it more technical. If x and y, are inversely proportional, then if one quantity doules, then the other halves.
Module 9: Further Numers and Equations However, the important part of this topic is what you do, which is almost always as follows: Write an equation that descries the situation. This will involve a constant. Use the information provided to find the value of. Now do whatever else the question requires. Writing the Equation We are often presented with a sentence involving the word proportional. Replace the phrase proportional to y an equals sign. Put the letter immediately after the equals sign. This means that multiplies the second quantity. Study the following tale. Sentence Equation y is proportional to x y x p is proportional to t p t y is proportional to the square of t y t V is proportional to the cue of r V r r is proportional to the square root of A r A If the sentence includes the phrase inversely proportional, then the right-hand side of the equation ecomes. Study the following tale. secon d qu an tity Sentence y is inversely proportional to x t is inversely proportional to q y is inversely proportional to the square of r P is inversely proportional to the cue root of r Equation y x t q y r P r Instead of a sentence, we are sometimes presented with an algeraic alternative involving the symol. This means proportional to. We replace this symol y an equals sign, and also insert the letter as efore. Study the following tale.
Lesson Statement using y x Equation y x x t x t A r A r f T f T y y x x F F r r f f m m Finding the value of Example We are usually given a pair of numers which we sustitute into the equation. With a it of rearrangement, it is possile to find the value of. Find the value of in each situation elow: Equation Information provided (a) y x y is 8 when x is 7. () r A r is when A is 6. (c) y is 9 when x is. y x (d) y is 7 when r is. y r (a) Sustitute the given values in the equation: 8 7. Rearrange: move the 7 to the left-hand side, changing the 8 multiplication to division:, so that is. 7 () Sustitute the given values in the equation: 6. But 6 8, so we have 8. Move the 8 to the other side:. We can now cancel to find the simplified value of. 8 6
Module 9: Further Numers and Equations (c) Sustitute the given values in the equation: 9. Now move the to the left-hand side, changing the division into multiplication: 9. So is 08. (d) Sustitute the given values in the equation: 7. But. Move this to the left-hand side:7. So is 7. The Full Routine Example We are now ready for the full routine required y IGCSE questions on this topic. Note that the same three stages are always required, ut that the question usually only mentions the third stage. (a) It is given that y is proportional to the square of x and that y is when x is 6. Find the value of y when x is 7. () It is given that R is inversely proportional to I, and that R is when I is 8. Find the value of R when I is. (c) It is given that P is inversely proportional to the square root of V, and that P is 6 when V is. Find the value of P when V is 00. (a) Write the equation: y x. Sustitute the nown values: 6. Rearrange:. If we now replace with 6, the equation ecomes: y x. We require the value of y when x is 7, so sustitute x 7 in the new version of the equation: y 7 96. () Write the equation: R. Sustitute the nown values:. I 8 Rearrange to give 8, so that we now now that is. The equation is therefore R. When I is, R is I given y R 6. (c) Write the equation: P. Sustitute the nown values: V 6. Rearrange to give: 6, so that must e 6. The equation is therefore P. When V is 00, V 7
Lesson the corresponding value of P is given y P. 00 0 Example It is given that A r and that A = 0.07 when r =.7. Use a calculator to find (correct to four significant figures): (a) the value of A when r is.0 () the value of r when A = 00. The equation is A r. Sustitute the nown values: 0.07 0.07.7. Rearrange to give.8....7 (a) Sustituting r =.0, the required value of A is.8.0 78. (correct to four significant figures). () We need to rearrange the equation suject. The new equation is and =.8, we otain four significant figures). A r to mae r the A r. Sustituting A = 00 00 r.6 (correct to.8 Activity 6. For each of the following statements, write down the corresponding equation: (a) h is proportional to m () E is proportional to the square of v (c) P is proportional to the square root of V (d) I is inversely proportional to W (e) E is inversely proportional to the square of d (f) q r (g) w s (h) a 8
Module 9: Further Numers and Equations Questions : without calculator. It is given that y is proportional to x and that y is 8 when x is. Find the value of y when x is. It is given that y is inversely proportional to x and that y is 6 when x is 7. Find the value of y when x is. It is given that f is proportional to the square root of T and that f is 0 when T is 00. Find the value of f when T is 6. It is given that F is inversely proportional to the square of r and that F is 8 when r is. Find the value of F when r is 0. Questions 6 7: with calculator 6 It is given that V r and that V = when r =.8. Find the value of V, correct to three significant figures, when r is.. 7 It is given that p, and that p is 7. when t is 8.. t Find the value of p when t is 0.0 (correct to three significant figures). Also find the value of t when p is 0.0 (correct to three significant figures). Activity 7 The volume V of a cylinder is directly proportional to the square of the ase radius r and the height h. (a) Write down an expression connecting V with r and h. () If h remains unchanged, ut r ecomes three times as long, how will V change? (c) How will V change if oth h and r are treled? 9
Lesson Activity 8 Here is an example from Astronomy: The time T for a planet to go round the Sun once is directly proportional to the cue of its distance (D) from the Sun. For the Earth, T = year and D is. 0 8 m. Calculate how long it taes the planet Mars to orit the Sun if its distance is. 0 8 m. Give your answer correct to significant figures. Suggested Answers to Activities Activity One (a) () (c) (d) (e) 6 (f) (g) 7 (h) 6 6 Activity Two (a) () (c) 0 0 0 Activity Three. 900 =,7.69 (.).,00 =,.6 (.06) Activity Four. Original % Increase or Multiplier Final 0
Module 9: Further Numers and Equations amount decrease? amount a 00.7 Increase.07 0.70 680.8 Decrease 0.968 6.6 c 79.6 Increase.6 89.79 d 000 8.7 Decrease 0.9 76.0 e 6.9.9 Increase.09 6. f 7.0. Decrease 0.97 709. g 70.68.8 Increase.08 7.80 h 8.66.6 Decrease 0.8 9.7. (a) 06.6 () 08.9. 07.0. 99.60. 99 Activity Five Fractional Index form 6 00 7 8 Root form Simplest possile form 6 00 0 7 8 (a) () 6 (c) (d) 6 (e) (f) 9 (a) () (c) 8
Lesson (d) 8 (e) 000 (f) 000 (a) 67 () 6 (c) 08 (d) 7 (e) (a) () (c) 9 (d) 9 9 6 (a) 7 () 7 (a) 6 () 0 6 6 6 Activity Six is a constant in each part (a) (h). (a) h m () (e) E d (f) q r (g) y =0 y = f = 7 f = 6 V = 68 7 p =.9 t = 6.9 ( s. f.) E v (c) P V (d) w s (h) I a W
Module 9: Further Numers and Equations Activity Seven (a) () V r h Since h remains unchanged, V will e proportional to the square of r. So if r is treled then V will ecome 9 times as large. (c) Since h is treled, V will ecome three times igger ecause of this. It will ecome 9 times igger ecause of the radius. So the total change is 9 = 7 Activity Eight T D So T = D For Earth: = (. 0 8 ) So = (. 0 8 ) = (.) 0 For Mars: T = (.) 0 (. 08 ) = (.) (.) =.6 years.